NCERT Solutions for Class 7 Mathematics Chapter 5 - Lines and Angles

Answer:

Yes, two acute angles are complementary to each other only if the sum of them is 90°.

Answer:

No, two obtuse angles cannot be complement of each other because their sum will be greater than 90°.

Answer:

No, two right angles can never be complement of each other, since their sum is also greater than 90°.

Answer:

Pairs (i) and (iv) are complementary.
Since in pair (i) 70° + 20° = 90° and in pair (iv) 35° + 55° = 90°

Answer:

(i) Complement of 45° = 90° – 45° = 45°
(ii) Complement of 65° = 90° – 65° = 25°
(iii) Complement of 41° = 90° – 41° = 49°
(iv) Complement of 54° = 90° – 54° = 36°

Answer:

Let the given angle be = x°
Therefore its complement = 90° – x°
Their difference = 12°
∴ x° – (90° – x°) = 12°
or, x° – 90° + x° = 12°
or, x° + x° = 12° + 90°
or, 2x° = 102°
or, x° = 102/2 = 51°
∴ First angle = 51°
Other angle = 90° – 51° = 39°
∴ The required angles are 51° and 39°.

Answer:

No, two obtuse angles cannot be supplementary because the sum of two obtuse angles would be more than 180°.

Answer:

No, two acute angles cannot be supplementary because the sum of two angles would be less than 180°.

Answer:

Yes, two right angles be supplementary because measure of each right angle is 90°.
∴ 90° + 90° = 180° = Supplementary

Answer:

The angles shown in figure (iii) are supplementary because in figure (iii) 130° + 50° = 180°.

Answer:

We know that the sum of two supplementary angles is 180°,
Therefore,
(i) Supplement of 100° = 180° – 100° = 80°
(ii) Supplement of 90° = 180° – 90° = 90°
(iii) Supplement of 55° = 180° – 55° = 125°
(iv) Supplement of 125° = 180° – 125° = 55°

Answer:

Let the smaller angle be = x°
Therefore, its supplement bigger angle = (x + 44)°
Since the sum of two supplement angles is 180°
∴ x° + (x + 44)° = 180°
or 2x° = 180° – 44° = 136°

Therefore, the smaller angle = 68°
and its supplement larger angle = 68° + 44° = 112°

Answer:

(i) In figure (i), angles $\angle$1 and $\angle$2 are adjacent angles because they have a common vertex and a common arm but no common interior points.
(ii) Angles in the figure (ii) are also adjacent angles.
(iii) Angles in the figure (iii) are not adjacent angles.
Reason: In figure (iii), angles $\angle$1 and $\angle$2 do not fulfil the first condition i.e. they have no common vertex. So, these angles are not adjacent angles.
(iv) Angles in the figure (iv) are not adjacent angles.
Reason: In figure (iv), angles $\angle$1 and $\angle$2 do not fulfil the condition that there are no common interior points.
(v) Angles in figure (v) are adjacent angles.

Answer:

(a) $\angle$AOB and $\angle$BOC are adjacent angles.
Reason: Because $\angle$AOB and $\angle$BOC have a common vertex O and a common arm OB. They do not have common interior points. (b) $\angle$BOD and $\angle$BOC are not adjacent angles. Reason: $\angle$BOD and $\angle$BOC have a common vertex O and a common arm OB. But they do not fulfil the third condition i.e. they have common interior point C.

Answer:

Yes, two adjacent angles can be supplementary if their sum is 180°.

Answer:

Yes, two adjacent angles can be complementary if their sum is 90°.

Answer:

Yes, two obtuse angles can be adjacent.

Answer:

Yes, an acute angle can be adjacent to an obtuse angle.

Answer:

No, two acute angles cannot form a linear pair.
Reason: The sum of two angles forming a linear pair is always 180°. But the sum of two acute angles is always less than 180°.

Answer:

No, two obtuse angles cannot form a linear pair.
Reason: The sum of two obtuse angle will be greater than 180°.

Answer:

Yes, two right angles can form a linear pair, because the sum of two right angles is 180°.

Answer:

Pairs (i) and (iv) will form a linear pair.
In fig. (i) 140° + 40° = 180°
In fig. (iv) 65° + 115° = 180°

Answer:

Since $\angle$1 and $\angle$2 form a linear pair
∴ $\angle$1 + $\angle$2 = 180°
∴ 30° + $\angle$2 = 180°
or $\angle$2 = 180° – 30° = 150°
Now, $\angle$3 and $\angle$1 form a pair of vertically opposite angles.
∴ $\angle$3 = $\angle$1
But $\angle$1 = 30°
∴ $\angle$3 = 30°

Answer:

Arms of the table stand forms a pair of vertically opposite angles.

Answer:

We know that the sum of complementary angles is 90°. Therefore,
(i) Complement of 20° = 90° – 20° = 70°
(ii) Complement of 63° = 90° – 63° = 27°
(iii) Complement of 57° = 90° – 57° = 33°

Answer:

We know that the sum of supplmentary angles is 180°.
Therefore,
(i) Supplement of 105° = 180° – 105° = 75°
(ii) Supplement of 87° = 180° – 87° = 93°
(iii) Supplement of 154° = 180° – 154° = 26°

Answer:

The pairs of angles in (ii), (v) and (vi) are complementary angles.
Because (ii) 63° + 27° = 90°, (v) 45° + 45° = 90°
and (vi) 80° + 10° = 90°
The pairs of angles in (i), (iii) and (iv) are supplementary angles.
Because (i) 65° + 115° = 180°, (ii) 112° + 68° =
180° and (iv) 130° + 50° = 180°.

Answer:

Let the angle be = x°
Therefore its complement = 90° – x°
Since the angle is equal to its complement
∴ x° = 90° – x°
or x° + x° = 90°
or 2x° = 90°
or x° = 90°/2
or x° = 45°
Therefore the required angle is 45°.

Answer:

Let the angle be x°
Therefore its supplement = 180° – x°
Since the angle is equal to its supplement
∴ x° = 180° – x°
or x° + x° = 180°
or 2x° = 180°
or x° = 180°/2
= 90°
Therefore, the required angle is 90°.

Answer:

If $\angle$1 is decreased, then $\angle$2 should be increased so that both the angles still remain supplementary.

Answer:

Since two angles be supplementary if sum of two angles is 180°.
(i) No; if two angles be acute (0° < q < 90°) then their sum cannot be equal to 180°.
(ii) No; if two angles be obtuse (90° < q < 180°) then their sum cannot be equal to 180°.
(iii) Yes; if two angles are right angle (= 90°) then their sum be 180°.

Answer:

Since the sum of two complementary angles is 90°.
So, if an angle is greater than 45°, then its complementary angle is less than 45°.

Answer:

(i) Yes, $\angle$1 is adjacent to $\angle$2.
(ii) No, $\angle$AOC is not adjacent to $\angle$AOE becuase they have common interior point C.
(iii) No, $\angle$COE and $\angle$COD do not form a linear pair.
(iv) Yes, $\angle$BOD and $\angle$DOA are supplementary because they form a straight angle i.e. 180°.
(v) Yes, $\angle$1 is vertically opposite to $\angle$4.
(vi) $\angle$BOC is the vertically opposite angle of $\angle$5.

Answer:

(i) $\angle$1, $\angle$4 and $\angle$5, $\angle$2 + $\angle$3 are vertically opposite angles.
(ii) $\angle$1, $\angle$5 and $\angle$4, $\angle$5 form a linear pair.

Answer:

No, $\angle$1 is not adjacent to $\angle$2.
Reason: $\angle$1 and $\angle$2 do not fullfil the first condition of adjacent angles i.e. they do not have a common vertex.

Answer:

In fig. (i) $\angle$x = 55° (Vertically opposite angles)
$\angle$y + 55° = 180° (Linear pair)
∴ $\angle$y = 180° – 55° = 125°
$\angle$y = $\angle$z (Vertically Opposite angles)
= 125°
In fig. (ii) $\angle$y + 40° = 180° (Linear pair)
∴ $\angle$y = 180° – 40° = 140°
$\angle$z = 40° (Vertically opposite angles)
$\angle$x + 25° + 40° = 180° (Linear pair)
∴ $\angle$x = 180° – 25° – 40°
= 115°

Answer:

90°

Answer:

180°

Answer:

adjacent angles and supplementary

Answer:

linear pair

Answer:

equal

Answer:

obtuse angles

Answer:

(i) A pair of obtuse vertically opposite angles are $\angle$AOD and $\angle$BOC.
(ii) Adjacent complementary angles are $\angle$AOB and $\angle$AOE.
(iii) Equal supplementary angles are $\angle$BOE and $\angle$EOD.
(iv) Unequal supplementary angles are $\angle$EOA ; $\angle$EOC
(v) Adjacent angles that do not form a linear pair are $\angle$AOB, $\angle$AOE; $\angle$AOE, $\angle$EOD; $\angle$EOD, $\angle$COD.

Answer:

Corners of the walls, sides of a box.

Answer:

Let us draw an equilateral triangle ABC with the help of three intersecting lines l, m and n intersecting each other at A, B and C respectively.

Answer:

Let us draw a rectangle PQRS with the help of four lines p, q, r and s interesecting each other at P, Q, R and S respectively. On measuring angles at the vertices P, Q, R and S respectively. We see that angles at the vertices P, Q, R and S are right angles.

Answer:

No, they should always not intersect in right angles.

Answer:

We can draw infinite number of transversals.

Answer:

When a line is a transversal to three lines, then there are three points of intersection.

Answer:

(i) A road crossing two or more roads.
(ii) A railway line crossing several other lines.

Answer:

$\angle$1 and $\angle$2 are pairs of corresponding angles.

Answer:

$\angle$3 and $\angle$4 are alternate interior angles.

Answer:

$\angle$5 and $\angle$6 are pairs of interior angles on the same side of the transversal.

Answer:

$\angle$7 and $\angle$8 are pairs of corresponding angles.

Answer:

$\angle$9 and $\angle$10 are alternate interior angles.

Answer:

$\angle$11 and $\angle$12 are exterior angles.

Answer:

l || m and t is a transversal
∴ $\angle$x = 60° (Alternate interior angles)

Answer:

a || b and c is a transvearsal.
∴ $\angle$y = 55° (Alternate interior angles)

Answer:

l₁ and l₂ is not parallel and t is a transversal.
∴ $\angle$1 $\ne$ $\angle$2

Answer:

lines l || m and t is a traversal
∴ $\angle$z + 60° = 180° .... (Pairs of interior angles on the same side of the transversal)
$\angle$z = 180° – 60° = 120°

Answer:

lines l || m and t is a traversal
∴ $\angle$x = 120°
.... (Pairs of corresponding angles)

Answer:

p || q and l is a transversal
∴ $\angle$a + 60° = 180°.... (Pairs of interior angles
on the same side of the transversal)
$\angle$a = 180° – 60°
$\angle$a = 120°
l || m and q is transversal
∴ $\angle$a = $\angle$1 ....(Pairs of corresponding angles)
o r $\angle$1 = 120°
$\angle$d = $\angle$1 = 120° .... (Vertically opposite
angles)
$\angle$1 + $\angle$c = 180° (Linear Pair)
120° + $\angle$c = 180°
$\angle$c = 180° – 120°
$\angle$c = 60°
$\angle$b = $\angle$c .... (Vertically opposite angles)
$\angle$b = 60°.

Answer:

Since alternate interior angles are equal.
∴ l || m

Answer:

Since corresponding angles are equal.
∴ $\angle$1 = 180° – 130° (linear pair)
$\angle$1 = 50°
∴ l || m.

Answer:

l || m.
∵ The sum of the interior angles on the same side of the transversal is supplementary.
∴ x + 70° = 180°
∴ x = 180° – 70° = 110°

Answer:

(i) a || b, then $\angle$1 = $\angle$5 If a transversal intersects two parallel lines then the corresponding angles are equal.
∴ By corresponding angle property, it is true.
(ii) If $\angle$4 = $\angle$6, then a|| b. If a transversal intersects two parallel lines then the alternate interior angles are equal.
∴ By alternate interior angle property it is true.
(iii) If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary.

Answer:

(i) Pairs of corresponding angles are:
$\angle$1, $\angle$5 ; $\angle$2, $\angle$6 ; $\angle$3, $\angle$7 ; $\angle$4, $\angle$8
(ii) Pairs of alternate interior angles are:
$\angle$2, $\angle$8 ; $\angle$3, $\angle$5
(iii) Pairs of interior angles on the same side of the transversal are:
$\angle$2, $\angle$5 ; $\angle$3, $\angle$8
(iv) Vertically opposite angles are:
$\angle$1, $\angle$3 ; $\angle$2, $\angle$4 ; $\angle$5, $\angle$7 ; $\angle$6, $\angle$8

Answer:

p || q and l is transversal
∴ e + 125° = 180° (Linear Pair)
e = 180° – 125°
e = 55°
$\angle$f = $\angle$e (vertically opposite angles)
$\angle$f = 55° (∵ $\angle$e = 55°)
$\angle$a = $\angle$e (Corresponding angles)
$\angle$a = 55° (∵ $\angle$e = 55°)
$\angle$c = $\angle$a (vertically opposite angles)
$\angle$c = 55° (∵ $\angle$a = 55°)
$\angle$d = 125° (Corresponding angles)
$\angle$b = $\angle$d (vertically opposite angles)
$\angle$b = 125° (∵ $\angle$d = 125°)
Hence, a = 55°, b = 125°, c = 55°, d = 125°, e = 55°, f = 55°

Answer:

(i) $\angle$1 + 110° = 180° (Linear Pair of angles)
$\angle$1 = 180° – 110°
$\angle$1 = 70°
But, $\angle$x = $\angle$1 (alternate angles)
$\angle$x = 70° (∵ $\angle$1 = 70°)
(ii) l || m and a is transversal
x = 100° (Corresponding angles)

Answer:

(i) AB || DE and BC is transversal
∴ $\angle$DGC = $\angle$ABC
(Corresponding angles)
But $\angle$ABC = 70° (Given)
∴ $\angle$DGC = 70°
(ii) BC || EF and DE is transversal
∴ $\angle$DEF = $\angle$DGC
(Corresponding angles)
But $\angle$DGC = 70°
∴ $\angle$DEF = 70°.

Answer:

(i) l is not parallel to m.
Reason: Q 126° + 44° = 170° ; which is not equal to 180°.
(ii) l is not parallel to m
Reason: Q $\angle$1 = 75° (Vertically opposite angles) and 75° + 75° = 150° ; which is not equal to 180°.
(iii) l is parallel to m.
Reason: Q $\angle$1 + 57° = 180°
(Linear Pair angles)
$\angle$1 = 180° – 57°
$\angle$1 = 123°
But these are alternate angles.
(iv) l is not parallel to m.
Reason: $\angle$1 = 72° (Vertically opposite angles)
and 72° + 98° = 170°; which is not equal to 180°.